Double angle in tension

S.1 \[\begin{aligned} \htmlClass{sym-phi}{\phi} &= 0.9 \quad \left(\text{Resistance factor | S16 Cl. 13.1}\right) \end{aligned}\]

S.2

S16 Cl. 3.2, E = 200 000 MPa

\[\begin{aligned} \htmlClass{sym-E}{E} &= 200\ \mathrm{GPa} \end{aligned}\]

S.3 \[\begin{aligned} \htmlClass{sym-b_0}{b_{0}} &= 102\ \mathrm{mm} \quad \left(\text{Width of flange or horizontal leg}\right) \end{aligned}\]

S.4 \[\begin{aligned} \htmlClass{sym-d_0}{d_{0}} &= 102\ \mathrm{mm} \quad \left(\text{Depth of section or height of vertical leg}\right) \end{aligned}\]

S.5 \[\begin{aligned} \htmlClass{sym-t}{t} &= 11.1\ \mathrm{mm} \quad \left(\text{Thickness of flange}\right) \end{aligned}\]

S.6 \[\begin{aligned} \htmlClass{sym-A_0}{A_{0}} &= 4280\ \mathrm{mm}^{2} \quad \left(\text{Cross-sectional area}\right) \end{aligned}\]

S.7 \[\begin{aligned} \htmlClass{sym-y_c}{y_{c}} &= 29.6\ \mathrm{mm} \quad \left(\text{Vertical distance between centroid and outside face of flange orhorizontal leg}\right) \end{aligned}\]

S.8 \[\begin{aligned} \htmlClass{sym-r_x}{r_{x}} &= 31.3\ \mathrm{mm} \quad \left(\text{Radius of gyration about axis XX}\right) \end{aligned}\]

S.9 \[\begin{aligned} \htmlClass{sym-r_y2}{r_{y2}} &= 46.6\ \mathrm{mm} \quad \left(\text{Radius of gyration about axis YY (10 mm back-to-back)}\right) \end{aligned}\]

S.10 \[\begin{aligned} \htmlClass{sym-r_z}{r_{z}} &= 20\ \mathrm{mm} \quad \left(\text{Minimum radius of gyration about minor principal axis(single angle)}\right) \end{aligned}\]

S.11

S16 Cl. 13.2, gross section yield

\[\begin{aligned} \htmlClass{sym-T_r}{T_{r}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-A_0}{4280\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-F_y}{300\ \mathrm{MPa}} \\ &= 1.156\ \mathrm{MN} \end{aligned}\]

S.12

S16 Cl. 10.4.1, L/r for a member in tension

\[\begin{aligned} \htmlClass{sym-lambda_x}{\lambda_{x}} &= \frac{\htmlClass{sym-L_x}{L_{x}}}{\htmlClass{sym-r_x}{r_{x}}} \\ &= \frac{\htmlClass{sym-L_x}{3000\ \mathrm{mm}}}{\htmlClass{sym-r_x}{31.3\ \mathrm{mm}}} \\ &= 95.85 \end{aligned}\]

S.13

S16 Cl. 10.4.1, L/r for a member in tension

\[\begin{aligned} \htmlClass{sym-lambda_y}{\lambda_{y}} &= \frac{\htmlClass{sym-L_y}{L_{y}}}{\htmlClass{sym-r_y2}{r_{y2}}} \\ &= \frac{\htmlClass{sym-L_y}{3000\ \mathrm{mm}}}{\htmlClass{sym-r_y2}{46.6\ \mathrm{mm}}} \\ &= 64.38 \end{aligned}\]

S.14

S16 Cl. 19.3.1, one angle between spacers

\[\begin{aligned} \htmlClass{sym-lambda_1}{\lambda_{1}} &= \frac{\max\left(\htmlClass{sym-L_x}{L_{x}}, \htmlClass{sym-L_y}{L_{y}}\right)}{\htmlClass{sym-n}{n} \cdot \htmlClass{sym-r_z}{r_{z}}} \\ &= \frac{\max\left(\htmlClass{sym-L_x}{3000\ \mathrm{mm}}, \htmlClass{sym-L_y}{3000\ \mathrm{mm}}\right)}{\htmlClass{sym-n}{3} \cdot \htmlClass{sym-r_z}{20\ \mathrm{mm}}} \\ &= 50 \end{aligned}\]

S.15 \[\begin{aligned} \htmlClass{sym-J}{J} &= 176000\ \mathrm{mm}^{4} \quad \left(\text{St-Venant torsional constant}\right) \end{aligned}\]

S.16 \[\begin{aligned} \htmlClass{sym-I_x}{I_{x}} &= 4.19 \times 10^{6}\ \mathrm{mm}^{4} \quad \left(\text{Moment of inertia about axis XX}\right) \end{aligned}\]

S.17 \[\begin{aligned} \htmlClass{sym-I_y}{I_{y}} &= \htmlClass{sym-r_y2}{r_{y2}}^{2} \cdot \htmlClass{sym-A_0}{A_{0}} \\ &= \left(\htmlClass{sym-r_y2}{46.6\ \mathrm{mm}}\right)^{2} \cdot \htmlClass{sym-A_0}{4280\ \mathrm{mm}^{2}} \\ &= 9.294 \times 10^{6}\ \mathrm{mm}^{4} \end{aligned}\]

S.18 \[\begin{aligned} \htmlClass{sym-e_v}{e_{v}} &= \left| \htmlClass{sym-g}{g} - \htmlClass{sym-y_c}{y_{c}} \right| \\ &= \left| \htmlClass{sym-g}{45\ \mathrm{mm}} - \htmlClass{sym-y_c}{29.6\ \mathrm{mm}} \right| \\ &= 15.4\ \mathrm{mm} \end{aligned}\]

S.19 \[\begin{aligned} \htmlClass{sym-M_fx}{M_{\mathrm{fx}}} &= \htmlClass{sym-e_v}{e_{v}} \cdot \htmlClass{sym-T_f}{T_{f}} \\ &= \htmlClass{sym-e_v}{15.4\ \mathrm{mm}} \cdot \htmlClass{sym-T_f}{50\ \mathrm{kN}} \\ &= 770\ \mathrm{N} \cdot \mathrm{m} \end{aligned}\]

S.20 \[\begin{aligned} \htmlClass{sym-stem}{\mathrm{stem}} &= 1 \quad \left(\text{Stem in tension}\right) \end{aligned}\]

S.21 \[\begin{aligned} \htmlClass{sym-S_xt}{S_{\mathrm{xt}}} &= 57800\ \mathrm{mm}^{3} \quad \left(\text{Elastic section modulus about axis XX}\right) \end{aligned}\]

S.22 \[\begin{aligned} \htmlClass{sym-S_xc}{S_{\mathrm{xc}}} &= \frac{\htmlClass{sym-I_x}{I_{x}}}{\htmlClass{sym-y_c}{y_{c}}} \\ &= \frac{\htmlClass{sym-I_x}{4.19 \times 10^{6}\ \mathrm{mm}^{4}}}{\htmlClass{sym-y_c}{29.6\ \mathrm{mm}}} \\ &= 141554\ \mathrm{mm}^{3} \end{aligned}\]

S.23

AISC 360-22 F9-3

\[\begin{aligned} \htmlClass{sym-M_y}{M_{y}} &= \htmlClass{sym-S_xt}{S_{\mathrm{xt}}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-S_xt}{57800\ \mathrm{mm}^{3}} \cdot \htmlClass{sym-F_y}{300\ \mathrm{MPa}} \\ &= 17.34\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.24

AISC 360-22 F9-1 and F9-2, M_p = 1.6 M_y, under F_y Z_x for every tabulated pair

\[\begin{aligned} \htmlClass{sym-M_n1}{M_{n1}} &= 1.6 \cdot \htmlClass{sym-M_y}{M_{y}} \\ &= 1.6 \cdot \htmlClass{sym-M_y}{17.34\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 27.74\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.25

AISC 360-22 F9-11

\[\begin{aligned} \htmlClass{sym-B}{B} &= 2.3 \cdot \frac{\htmlClass{sym-d_0}{d_{0}}}{\htmlClass{sym-L_y}{L_{y}}} \cdot \sqrt{\frac{\htmlClass{sym-I_y}{I_{y}}}{\htmlClass{sym-J}{J}}} \\ &= 2.3 \cdot \frac{\htmlClass{sym-d_0}{102\ \mathrm{mm}}}{\htmlClass{sym-L_y}{3000\ \mathrm{mm}}} \cdot \sqrt{\frac{\htmlClass{sym-I_y}{9.294 \times 10^{6}\ \mathrm{mm}^{4}}}{\htmlClass{sym-J}{176000\ \mathrm{mm}^{4}}}} \\ &= 0.5683 \end{aligned}\]

S.26

AISC 360-22 F9-10

\[\begin{aligned} \htmlClass{sym-M_cr}{M_{\mathrm{cr}}} &= 1.95 \cdot \frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-L_y}{L_{y}}} \cdot \sqrt{\htmlClass{sym-I_y}{I_{y}} \cdot \htmlClass{sym-J}{J}} \cdot \left(\htmlClass{sym-B}{B} + \sqrt{1 + \htmlClass{sym-B}{B}^{2}}\right) \\ &= 1.95 \cdot \frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-L_y}{3000\ \mathrm{mm}}} \cdot \sqrt{\htmlClass{sym-I_y}{9.294 \times 10^{6}\ \mathrm{mm}^{4}} \cdot \htmlClass{sym-J}{176000\ \mathrm{mm}^{4}}} \cdot \left(\htmlClass{sym-B}{0.5683} + \sqrt{1 + \htmlClass{sym-B}{0.5683}^{2}}\right) \\ &= 285.7\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.27

AISC 360-22 F9-8

\[\begin{aligned} \htmlClass{sym-L_p}{L_{p}} &= 1.76 \cdot \htmlClass{sym-r_y2}{r_{y2}} \cdot \sqrt{\frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}}} \\ &= 1.76 \cdot \htmlClass{sym-r_y2}{46.6\ \mathrm{mm}} \cdot \sqrt{\frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{300\ \mathrm{MPa}}}} \\ &= 2.118\ \mathrm{m} \end{aligned}\]

S.28

AISC 360-22 F9-9

\[\begin{aligned} \htmlClass{sym-L_r}{L_{r}} &= \frac{1.95 \cdot \frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}} \cdot \sqrt{\htmlClass{sym-I_y}{I_{y}} \cdot \htmlClass{sym-J}{J}}}{\htmlClass{sym-S_xt}{S_{\mathrm{xt}}}} \cdot \sqrt{\frac{2.36 \cdot \frac{\htmlClass{sym-F_y}{F_{y}}}{\htmlClass{sym-E}{E}} \cdot \htmlClass{sym-d_0}{d_{0}} \cdot \htmlClass{sym-S_xt}{S_{\mathrm{xt}}}}{\htmlClass{sym-J}{J}} + 1} \\ &= \frac{1.95 \cdot \frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{300\ \mathrm{MPa}}} \cdot \sqrt{\htmlClass{sym-I_y}{9.294 \times 10^{6}\ \mathrm{mm}^{4}} \cdot \htmlClass{sym-J}{176000\ \mathrm{mm}^{4}}}}{\htmlClass{sym-S_xt}{57800\ \mathrm{mm}^{3}}} \cdot \sqrt{\frac{2.36 \cdot \frac{\htmlClass{sym-F_y}{300\ \mathrm{MPa}}}{\htmlClass{sym-E}{200\ \mathrm{GPa}}} \cdot \htmlClass{sym-d_0}{102\ \mathrm{mm}} \cdot \htmlClass{sym-S_xt}{57800\ \mathrm{mm}^{3}}}{\htmlClass{sym-J}{176000\ \mathrm{mm}^{4}}} + 1} \\ &= 30.42\ \mathrm{m} \end{aligned}\]

S.29

AISC 360-22 F9-6

\[\begin{aligned} \htmlClass{sym-M_n2}{M_{n2}} &= \htmlClass{sym-M_n1}{M_{n1}} - \frac{\left(\htmlClass{sym-M_n1}{M_{n1}} - \htmlClass{sym-M_y}{M_{y}}\right) \cdot \left(\htmlClass{sym-L_y}{L_{y}} - \htmlClass{sym-L_p}{L_{p}}\right)}{\htmlClass{sym-L_r}{L_{r}} - \htmlClass{sym-L_p}{L_{p}}} \\ &= \htmlClass{sym-M_n1}{27.74\ \mathrm{kN} \cdot \mathrm{m}} - \frac{\left(\htmlClass{sym-M_n1}{27.74\ \mathrm{kN} \cdot \mathrm{m}} - \htmlClass{sym-M_y}{17.34\ \mathrm{kN} \cdot \mathrm{m}}\right) \cdot \left(\htmlClass{sym-L_y}{3000\ \mathrm{mm}} - \htmlClass{sym-L_p}{2.118\ \mathrm{m}}\right)}{\htmlClass{sym-L_r}{30.42\ \mathrm{m}} - \htmlClass{sym-L_p}{2.118\ \mathrm{m}}} \\ &= 27.42\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.30 \[\begin{aligned} \htmlClass{sym-lambda_l}{\lambda_{l}} &= \frac{\htmlClass{sym-b_0}{b_{0}}}{\htmlClass{sym-t}{t}} \\ &= \frac{\htmlClass{sym-b_0}{102\ \mathrm{mm}}}{\htmlClass{sym-t}{11.1\ \mathrm{mm}}} \\ &= 9.189 \end{aligned}\]

S.31

AISC 360-22 Table B4.1b case 12, legs of angles

\[\begin{aligned} \htmlClass{sym-lambda_p}{\lambda_{p}} &= 0.54 \cdot \sqrt{\frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}}} \\ &= 0.54 \cdot \sqrt{\frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{300\ \mathrm{MPa}}}} \\ &= 13.94 \end{aligned}\]

S.32

AISC 360-22 Table B4.1b case 12, legs of angles

\[\begin{aligned} \htmlClass{sym-lambda_r}{\lambda_{r}} &= 0.91 \cdot \sqrt{\frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}}} \\ &= 0.91 \cdot \sqrt{\frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{300\ \mathrm{MPa}}}} \\ &= 23.5 \end{aligned}\]

S.33

AISC 360-22 F10.3(a), compact leg: does not apply

\[\begin{aligned} \htmlClass{sym-M_n3}{M_{n3}} &= 1 \cdot \htmlClass{sym-M_n1}{M_{n1}} \\ &= 1 \cdot \htmlClass{sym-M_n1}{27.74\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 27.74\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.34

AISC 360-22 F9, the lowest of the limit states

\[\begin{aligned} \htmlClass{sym-M_nx}{M_{\mathrm{nx}}} &= \min\left(\htmlClass{sym-M_n1}{M_{n1}}, \htmlClass{sym-M_n2}{M_{n2}}, \htmlClass{sym-M_n3}{M_{n3}}\right) \\ &= \min\left(\htmlClass{sym-M_n1}{27.74\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-M_n2}{27.42\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-M_n3}{27.74\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 27.42\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.35

S16 Cl. 13.1 a), phi on the nominal strength

\[\begin{aligned} \htmlClass{sym-M_rx}{M_{\mathrm{rx}}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-M_nx}{M_{\mathrm{nx}}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-M_nx}{27.42\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 24.68\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.36

S16 Cl. 13.9.1

\[\begin{aligned} \htmlClass{sym-I_a}{I_{a}} &= \frac{\htmlClass{sym-T_f}{T_{f}}}{\htmlClass{sym-T_r}{T_{r}}} + \frac{\htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}} \\ &= \frac{\htmlClass{sym-T_f}{50\ \mathrm{kN}}}{\htmlClass{sym-T_r}{1.156\ \mathrm{MN}}} + \frac{\htmlClass{sym-M_fx}{770\ \mathrm{N} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{24.68\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.07447 \end{aligned}\]

S.37

S16 Cl. 13.9.3 b), S at the compression fibre

\[\begin{aligned} \htmlClass{sym-I_b}{I_{b}} &= \frac{\htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}} - \frac{\htmlClass{sym-T_f}{T_{f}} \cdot \htmlClass{sym-S_xc}{S_{\mathrm{xc}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}} \cdot \htmlClass{sym-A_0}{A_{0}}} \\ &= \frac{\htmlClass{sym-M_fx}{770\ \mathrm{N} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{24.68\ \mathrm{kN} \cdot \mathrm{m}}} - \frac{\htmlClass{sym-T_f}{50\ \mathrm{kN}} \cdot \htmlClass{sym-S_xc}{141554\ \mathrm{mm}^{3}}}{\htmlClass{sym-M_rx}{24.68\ \mathrm{kN} \cdot \mathrm{m}} \cdot \htmlClass{sym-A_0}{4280\ \mathrm{mm}^{2}}} \\ &= -0.03581 \end{aligned}\]